LINEARIZED STABILITY PRINCIPLE FOR DIFFERENTIAL EQUATIONS WITH DELAYS

2017. Т. 17, No 3. С. 129–134 129


Introduction
Many applied scientists ask the following question: why do we need to study differential equations with delays when so much is known about equations without delays and these equations are much easier? The answer is: because so many processes, both natural and man-made, in biology, medicine, chemistry, engineering, economics etc. involve time delays. Whether you like it or not, time delays occur so often in almost every situation that to ignore them means to ignore the reality.
A simple example in nature is reforestation. After cutting a forest and replanting, it will take at least 20 years before reaching any kind of maturity. Hence any mathematical model of forest harvesting should involve time delays.
Similarly any model of population dynamics without delays is an approximation at best. Another motivation for studying DDE comes from their applications in feedback control theory. In his study of ship stabilization Minorsky (1942) pointed out very clearly the importance of incorporating delays in the feedback mechanism.
We would like to present here several applied models with time delays.  To find a solution of this equation we need to assign an initial function instead of an initial value. Suppose that we have the following initial function ( ) = 1; ∈ [−2; 0]. There is no exact formula for the solution of this problem, but we can find this solution by the method of steps.
So the behavior of solutions for equations with delay and without delay is quite different. Actually the behavior of solutions also depends on the value of delay. In particular, consider the equation with any constant delay ( ) = − ( − ).
If ≤ , then the solution of this equation with the same initial function as above is positive, monotone and tends to zero. But there are also oscillating solutions of this equation.
If ≥ ≥ , then all solutions for any initial function oscillate and tend to zero.
If ≥ , then all solutions oscillate and the equation is unstable.
There are several important questions on the asymptotic behavior of solutions of delay differential equations. Here we consider only one property which is a global stability of the equation. What does it mean?
Consider the nondelay logistic equation The carrying capacity K is a positive equilibrium. It is a simple fact that for ( ) > > 0 this equilibrium is locally asymptotically stable. But what is more interesting, this solution attracts all positive solutions. In this case we say that ( ) = is a global attractor or that the equation is globally asymptotically stable.
We assume that a global solution of (2)-(3) exists and is unique.
The following stability definition is for nonlinear equation (2); it also is applies to linear equation (4).
The following lemmas are modifications of the results of paper [2].
Then all solutions of equation (4) are bounded.

Main result
Now we can formulate our main result. To this end consider the following condition. There exists a partition {1,2, … , } = ∪ ∪ , and indices i k , such that 2

Examples
In this section we will apply Theorem 1 to obtain global stability results for some nonlinear delay equations. It means that we will prove that all solutions or all positive solutions tend to an equilibrium as t → ∞.
Such applications consist of two steps. First by Theorem 1, Part 1 we prove that all solutions of the equation are bounded. Then by the second part of Theorem 1 we prove that every bounded solution tends to zero or to another equilibrium. Example 1. Consider the equation where 0 ≤ | | < , lim →∞ sup − ( ) < ∞.
We have only one nonlinear term.
Consider now the linear equation where b(t) ≥ b > 0. There exists λ, 0 < λ < 1, such that |a| < λb ≤ λb(t). By Lemma 4 equation (20) is asymptotically stable. Hence all solutions of this equation are bounded. If we take α(t) = b, then condition (13) is satisfied. By the first part of Theorem 1 all solutions of equation (19) are bounded.
If we take again α(t) = b and two arbitrary numbers A, B, A < B as a lower and an upper bounds, respectively, then condition (13) is satisfied for any A ≤ u i ≤ B, i = 1, 2. Hence by the second part of Theorem 1 for any solution of equation (19) such that A ≤ x(t) ≤ B we have lim →∞ ( ) = 0. Since A and B are arbitrary numbers, then for any solution of equation (19) we have lim →∞ ( ) = 0.
Consider now the linear equation